Mistakes give rise to Problems- 16

You can remove the brackets in a fraction (note that this is fraction, not the Legendre symbol) , just like in the following fraction (ab)=ab\displaystyle \biggl(\color{#3D99F6}{\dfrac{a}{b}} \biggr) = \color{#3D99F6}{\dfrac{a}{b}}


But if you do that in the Binomial Coefficient\color{#69047E}{\textbf{Binomial Coefficient}}, like (ab)=ab\displaystyle \dbinom{a}{b} = \dfrac{a}{b} then it's a Big Mistake !!!\color{#D61F06}{\textbf{Big Mistake !!!}}


But for how many ordered pairs (a,b)\color{#20A900}{(a,b)} such that 0a200\leq a \leq 20 and 0b200\leq b \leq 20 is the above said "false" property seen to be "true" ?


Details and assumptions

\bullet\quad Factorial (a!a!) is defined only for non-negative integers.

\bullet\quad(ab)=a!b!×(ab)!\dbinom{a}{b} = \dfrac{a!}{b! \times (a-b)!}

\bullet\quad(ab)=0\dbinom{a}{b} = 0 if a<ba<b.

\bullet\quad0!=10! =1


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